#474040

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(27,33)(28,35)(30,37)(32,38)(34,39)(36,40);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,20,23)(18,21,22,24)(25,26,29,31)(27,34,36,30)(28,32,35,38)(33,39,40,37);(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(25,27,35,40,29,36,28,33)(26,30,38,39,31,34,32,37);(11,15)(14,16)(18,22)(21,24)(26,31)(30,34)(32,38)(37,39)

Polycyclic presentation:

2^9|a^2=egi,b^2=egh,c^2=fi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381236);